Printable primary math worksheet for math grades 1 to 6 based on ... - Free Printable

Let's solve each of these square root estimation problems step by step. The goal is to estimate the square root of a number by finding two consecutive perfect squares between which the number lies. Then, we can say the square root lies between the square roots of those perfect squares.

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🔹 Step-by-step method:

1. Find two perfect squares such that:
$$
a^2 < \text{number} < b^2
$$
2. Then:
$$
\sqrt{a^2} < \sqrt{\text{number}} < \sqrt{b^2} \Rightarrow a < \sqrt{\text{number}} < b
$$

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Now let’s go through each problem:

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1. $\sqrt{200}$ is between 14 and 15

✔ Already given:
$14^2 = 196$, $15^2 = 225$ → $196 < 200 < 225$ → So $\sqrt{200}$ is between 14 and 15

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2. $\sqrt{37}$ is between ___ and ___

- $6^2 = 36$, $7^2 = 49$ → $36 < 37 < 49$
→ So $\sqrt{37}$ is between 6 and 7

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3. $\sqrt{31}$ is between ___ and ___

- $5^2 = 25$, $6^2 = 36$ → $25 < 31 < 36$
→ So $\sqrt{31}$ is between 5 and 6

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4. $\sqrt{2}$ is between ___ and ___

- $1^2 = 1$, $2^2 = 4$ → $1 < 2 < 4$
→ So $\sqrt{2}$ is between 1 and 2

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5. $\sqrt{134}$ is between ___ and ___

- $11^2 = 121$, $12^2 = 144$ → $121 < 134 < 144$
→ So $\sqrt{134}$ is between 11 and 12

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6. $\sqrt{350}$ is between ___ and ___

- $18^2 = 324$, $19^2 = 361$ → $324 < 350 < 361$
→ So $\sqrt{350}$ is between 18 and 19

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7. $\sqrt{8}$ is between ___ and ___

- $2^2 = 4$, $3^2 = 9$ → $4 < 8 < 9$
→ So $\sqrt{8}$ is between 2 and 3

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8. $\sqrt{500}$ is between ___ and ___

- $22^2 = 484$, $23^2 = 529$ → $484 < 500 < 529$
→ So $\sqrt{500}$ is between 22 and 23

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9. $\sqrt{80}$ is between ___ and ___

- $8^2 = 64$, $9^2 = 81$ → $64 < 80 < 81$
→ So $\sqrt{80}$ is between 8 and 9

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10. $\sqrt{60}$ is between ___ and ___

- $7^2 = 49$, $8^2 = 64$ → $49 < 60 < 64$
→ So $\sqrt{60}$ is between 7 and 8

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11. $\sqrt{111}$ is between ___ and ___

- $10^2 = 100$, $11^2 = 121$ → $100 < 111 < 121$
→ So $\sqrt{111}$ is between 10 and 11

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12. $\sqrt{230}$ is between ___ and ___

- $15^2 = 225$, $16^2 = 256$ → $225 < 230 < 256$
→ So $\sqrt{230}$ is between 15 and 16

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13. $\sqrt{320}$ is between ___ and ___

- $17^2 = 289$, $18^2 = 324$ → $289 < 320 < 324$
→ So $\sqrt{320}$ is between 17 and 18

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14. $\sqrt{190}$ is between ___ and ___

- $13^2 = 169$, $14^2 = 196$ → $169 < 190 < 196$
→ So $\sqrt{190}$ is between 13 and 14

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15. $\sqrt{3}$ is between ___ and ___

- $1^2 = 1$, $2^2 = 4$ → $1 < 3 < 4$
→ So $\sqrt{3}$ is between 1 and 2

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16. $\sqrt{5}$ is between ___ and ___

- $2^2 = 4$, $3^2 = 9$ → $4 < 5 < 9$
→ So $\sqrt{5}$ is between 2 and 3

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17. $\sqrt{150}$ is between ___ and ___

- $12^2 = 144$, $13^2 = 169$ → $144 < 150 < 169$
→ So $\sqrt{150}$ is between 12 and 13

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18. $\sqrt{115}$ is between ___ and ___

- $10^2 = 100$, $11^2 = 121$ → $100 < 115 < 121$
→ So $\sqrt{115}$ is between 10 and 11

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19. $\sqrt{70}$ is between ___ and ___

- $8^2 = 64$, $9^2 = 81$ → $64 < 70 < 81$
→ So $\sqrt{70}$ is between 8 and 9

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20. $\sqrt{90}$ is between ___ and ___

- $9^2 = 81$, $10^2 = 100$ → $81 < 90 < 100$
→ So $\sqrt{90}$ is between 9 and 10

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✔ Final Answer Key (filled in):

| Problem | Between |
|--------|---------|
| $\sqrt{200}$ | 14 and 15 |
| $\sqrt{37}$ | 6 and 7 |
| $\sqrt{31}$ | 5 and 6 |
| $\sqrt{2}$ | 1 and 2 |
| $\sqrt{134}$ | 11 and 12 |
| $\sqrt{350}$ | 18 and 19 |
| $\sqrt{8}$ | 2 and 3 |
| $\sqrt{500}$ | 22 and 23 |
| $\sqrt{80}$ | 8 and 9 |
| $\sqrt{60}$ | 7 and 8 |
| $\sqrt{111}$ | 10 and 11 |
| $\sqrt{230}$ | 15 and 16 |
| $\sqrt{320}$ | 17 and 18 |
| $\sqrt{190}$ | 13 and 14 |
| $\sqrt{3}$ | 1 and 2 |
| $\sqrt{5}$ | 2 and 3 |
| $\sqrt{150}$ | 12 and 13 |
| $\sqrt{115}$ | 10 and 11 |
| $\sqrt{70}$ | 8 and 9 |
| $\sqrt{90}$ | 9 and 10 |

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📌 Summary:

This exercise helps build estimation skills for irrational square roots. You don’t need a calculator — just know your perfect squares up to around $25^2 = 625$.

For example:
- $1^2 = 1$
- $2^2 = 4$
- $3^2 = 9$
- ...
- $10^2 = 100$
- $15^2 = 225$
- $20^2 = 400$
- $25^2 = 625$

With this knowledge, you can quickly estimate any square root!

Let me know if you'd like to refine the estimates (e.g., closer approximations).